scholarly journals The domain derivative for semilinear elliptic inverse obstacle problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Frank Hettlich
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Obstacle Problem ◽  
Obstacle Problems ◽  
Iterative Regularization ◽  
Regularization Scheme ◽  
Numerical Examples ◽  
Semilinear Elliptic ◽  
Domain Derivative

<p style='text-indent:20px;'>We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.</p>

scholarly journals Maximum principles for a class of semilinear elliptic boundary-value problems

1995 ◽  
Vol 52 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Zhang Hailiang
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Elliptic Boundary ◽  
Maximum Principles ◽  
Robin Boundary Conditions ◽  
Elliptic Boundary Value ◽  
Physical Problems ◽  
Semilinear Elliptic ◽  
Robin Boundary

For years it has remained a problem to find suitable functionals satisfying certain maximum principles for solutions of the equation Δu + f(x, u) = 0. In this paper, maximum principles for certain functionals which are defined on solutions of semilinear elliptic equations subject to mixed or Robin boundary conditions are obtained. The principles derived may be used to deduce bounds on important quantities in physical problems of interest.

Points of Spherical Maxima and Solvability of Semilinear Elliptic Equations

1991 ◽  
Vol 43 (4) ◽  
pp. 825-831 ◽  
Author(s):  
Martin Schechter ◽  
Kyril Tintarev
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equations ◽  
Nonlinear Functional ◽  
Semilinear Elliptic

AbstractWe give mild sufficient conditions on a nonlinear functional to have eigenvalues. These results are intended for the study of boundary value problems for semilinear elliptic equations.

The Landweber Iteration for an Inverse Scattering Problem

1995 ◽  
Author(s):  
Martin Hanke ◽  
Frank Hettlich ◽  
Otmar Scherzer
Keyword(s):  
Numerical Solution ◽  
Inverse Scattering ◽  
Obstacle Problem ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Operator ◽  
Iteration Scheme ◽  
Far Field ◽  
Numerical Examples

Abstract A Landweber iteration scheme is presented for the numerical solution of an inverse obstacle problem. The method uses a recently obtained characterization of the Fréchet derivative of the far field operator and its adjoint. The performance of the method is illustrated by some numerical examples. Some theoretical aspects are pointed out to motivate the use of nonlinear Landweber iteration.

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